Facing a 15-point halftime deficit, Couch threw a pair of short touchdown passes and a wild two-point conversion pass in the third quarter and helped set up Phil Dawson for the decisive 35-yard field goal with 4 1/2 minutes to play.
Dawson remained perfect inside 40 yards since the opening game last season as the Browns (4-4) stayed in the thick of the playoff hunt, leaving Giants Stadium a half game behind Pittsburgh for first place in the AFC North.
"Somebody is smiling down on us," Coach Butch Davis said. "We made a great play that ends the ballgame."
"This was a huge win, not only for myself but for my team and our hunt to get back into the playoff race," said Couch, who completed 33 of 49 passes for 307 yards. "We needed this one badly. It wasn't easy and it wasn't pretty."
It was ugly for the Jets (2-5), who built a 21-3 lead 37 seconds into the second quarter but suffered a devastating loss, their fifth in six games.
The Jets self-destructed in the second half but still had a chance to send the game to overtime when Chad Pennington drove his team to Cleveland's 26. But Courtney Brown and Alvin McKinley blocked John Hall's 43-yard attempt.
"I heard two thuds and I knew it was blocked," Hall said.
The ball hit Brown and McKinley before falling to the ground, possibly taking the Jets' season with it. Considered a playoff contender at the start of the year, they began the day last in the AFC East.
"It's just gut-wrenching," said Pennington, who was 19 of 26 for 183 yards with two touchdowns and an interception.
Things looked good early for the Jets, who opened the scoring on Santana Moss' 63-yard punt return 39 seconds into the game.
Pennington then threw a 37-yard touchdown pass to Laveranues Coles and an eight-yarder to Wayne Chrebet in a span of 6 1/2 minutes.
The Jets had a 21-6 lead at halftime, but it was a different story over the final 30 minutes. They lost eight yards on their next four possessions and totaled only 37 yards in the second half.
"We self-destructed in the second half," Coach Herman Edwards said. "It's all on us."
The Browns got within 21-13 when Couch found tight end Mark Campbell from five yards with 4:40 left in the third quarter, finishing an 11-play, 60-yard drive.
Later in the quarter, Gerald Warren popped the ball loose while sacking Pennington and Brown recovered at New York's 11. Couch ended up finding rookie Andre Davis from four yards to get the Browns within 21-19.
Davis decided to go for two and the tie. Steve White got pressure on Couch, who heaved the ball to the end zone while falling to his back and watched Dennis Northcutt make the catch between a pair of defenders.
"We needed a two-point conversion and I was getting sacked," Couch said. "I actually saw a white jersey flash across my eyes and just threw it up and let Dennis make a play."
After the fifth of six straight empty possessions by the Jets, Couch marched his team to New York's 18 on 13 plays before Dawson kicked the decisive field goal.
"Two weeks ago, it was a 10-game playoff schedule," Davis said. "It's getting to that point in the season. Hopefully, we found a little rhythm."